Donald Trump claimed victory Tuesday following his first debate against Hillary Clinton, saying the showdown made it clear that she is “the candidate of yesterday” and he is the candidate of “the future.”

Making a campaign swing through Florida, Mr. Trump said the campaign is already reaping the benefits, raising more than $18 million in a post-debate fundraising push.

“For 90 minutes on issue after issue Hillary Clinton defended the terrible status quo, while I laid out our plan, all of us together, to bring jobs, security and prosperity back to the American people,” Mr. Trump said at a campaign stop in Melbourne, Florida. “For 90 minutes she argued against change, while I called for dramatic change.”

At the same time, debate moderator Lester Holt of NBC News faced criticism for challenging some of the Republican nominee’s statements, while failing to press Mrs. Clinton with follow-up questions.

“The single weapon she has got in the media,” Mr. Trump said Tuesday. “Without the mainstream media, she wouldn’t even be here, folks, that I can tell you. She wouldn’t even be here. She wouldn’t have a chance.”

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Still, political pundits from across the ideological spectrum said the billionaire businessman could have done more to capitalize on the momentum that he carried into the debate, with polls showing that he had cut into Mrs. Clinton’s lead nationally and in key swing states.

Mr. Trump, meanwhile, declared victory, pointing to online surveys that showed respondents believed he won the first of three scheduled debates.

He also outlined parts of his agenda that went unmentioned in the debate — including his plan to build a wall along the U.S.-Mexico border, to expand school choice, and nominate justices to the Supreme Court that will uphold the Constitution. He called Mrs. Clinton “Crooked Hillary” — something he did not do in the debate — a number of times.

And he said he took it easy on Mrs. Clinton by not bringing up her husband Bill Clinton’s extramarital affairs.

“I didn’t want to do anything to embarrass her,” he said. “But I watched her, and she was stuck in the past.”